Simple Occluder Scene

We analyze the gradients and Hessians for a scene with a bright area emitter was placed over a large plane, with two occluders located between the light and the plane at different distances (see Figure 5 of the paper). The first three comparisons show a visualization of the first derivative of the irradiance (its gradient) across the bottom plane, using the mapping (dE/dx,dE/dy) → (r,g). We compare the method by Ward et al. and our approach with increasing numbers of hemispherical samples to an analytic ground truth. The last comparison shows the eigenvalues of the irradiance Hessian using the mapping (λ₁, λ₂,λ₃) → (r,g,b). Here we compare the method by Jarosz et al. and our approach to an analytic ground truth.

Convergence

Below we compare the behavior of our Occlusion Hessian method to the Bounded Split-Sphere heuristic with increasing cache point densities. All images used 4096 gather rays, and the Split-Sphere used a maximum radius of 150px (a tighter bound was not possible for very limited cache point counts). Note how the Occlusion Hessian produces a high-quality result even with 500 cache records, while the Split-Sphere suffers from large artifacts even with 1000 records. These artifacts occur because the strong constraint on the record count requires setting such a high error threshold that cache records from the ceiling are used on the walls, and vice-versa.

San Miguel

An interior of San Miguel with strong indirect illumination. Note the areas behind the paintings where the Occlusion Hessian is able to resolve the shadows, while the Split-Sphere heuristic (clamped to 20px and the gradient) largely misses these important features.

Robustness (Radiometric vs. Occlusion Hessian)

Below we compare the Radiometric Hessian [Jarosz et al. 2012] to our Occlusion Hessian method and also incorporate our improvements for relative error and robustness. The first comparison shows the result when using an absolute error, while the second uses a relative measure. The third adds 1% of the total irradiance to all the triangle radiances prior to computing the Hessian. While this improves the result for the Radiometric Hessian method, it still retains a few distracting artifacts, that are not present for the Occlusion Hessian.

Robustness to Gather Ray Count

Our method is robust to low gather ray counts, producing nearly identical cache point distributions with 256 (left) as with 4096 (right) gather rays per cache record.

Robustness to Scene Illumination

Irradiance rendering of the Cornell Box after changing the back wall so it has an albedo of 0. The Geometric Hessian does not take the radiometry of the scene into account, so its cache point distribution is the same as for the standard Cornell Box, which is sub-optimal. Our new Occlusion Hessian adapts its distribution to match the radiometry in this new configuration, producing a higher quality result. Note that we have scaled the difference image color range to ease comparison.

Sponza

We compare our error metric to several previous approaches in the Sponza atrium scene.

Comparison to Split-Sphere

In Sponza, our Occlusion Hessian is superior to the Split-Sphere in both defining shadow details and eliminating interpolation artifacts across surfaces. The shadows at the base on the column are captured by the Occlusion Hessian, while the Split-Sphere heuristic shows severe interpolation artifacts, even when bounded to 20px and clamped by the gradient.

Here we first show the final rendered images and the RMS error, comparing a path traced ground truth (top-left) to our approach (top-right) to the Split-Sphere, both the pure variant (bottom-left) and bounded to 20px and clamped by the gradient (bottom-right).

Next, we show a direct visualization of the radii and footprints of the cache records for these three error metrics. The Pure Split-Sphere is unable to detect the indirect shadows on the columns, while the Bounded Split-Sphere (clamped to 20px and the gradient) as well as our Occlusion Hessian correctly predict smaller radii in these rapidly changing regions. Unfortunately, the contrast in the Split-Sphere is too high, leading to lost detail due to the greedy nature of the irradiance caching algorithm.

Comparison to Jarosz et al. 2012

We also compare to the Anisotropic (occlusion-unaware) Hessian approaches from Jarosz et al. 2012. These methods ignore visibility changes, which are important for detecting rapidly changing regions. Hence, in these direct visualization of the radii and footprints of the cache records, both previous methods are unable to detect the indirect shadows on the columns, while our Occlusion Hessian can correctly predict smaller radii.

Comparison to Jarosz et al. 2012

Visualization of the radii estimated for the Occlusion Hessian using both absolute and relative measures for the total error. A relative measure allows larger radii in bright regions where absolute differences in irradiance induce lower perceived error.

Isotropic vs. Anisotropic Metric

Comparison of isotropic to anisotropic cache records for the relative Occlusion Hessian metric with the same threshold. The eccentricity of the cache records is visualized as the filled-in color, with green representing isotropic records and dark blue representing maximum anisotropy. Note that we have clamped the major axis at twice the length of the minor axis. The anisotropic metric adapts the eccentricity to the local irradiance curvature, allowing for less cache records for the same error threshold.